Subgroups — #13068596

Order 1408.

400 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₁₁⋊C₈)	C₂	1	2	yes	maximal
3	C₂×C₄⋊(C₁₁⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#10622162	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×C₁₁	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₁₁	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₁	—	2	4	no	—
9	C₂×C₄²×C₁₁	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₁₁	C₄	1	1	yes	characteristic
11	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
13	C₂²×C₁₁⋊C₈	C₂²	1	4	yes	—
14	C₂²⋊(C₁₁⋊C₈)	—	2	4	no	—
15	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
16	C₂⁴×C₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₁	—	2	2	no	—
18	C₂²×C₄×C₁₁	—	2	2	no	—
19	C₂²×C₄×C₁₁	—	2	2	no	—
20	C₂²×C₄×C₁₁	—	2	2	no	—
21	C₂²×C₄×C₁₁	—	2	2	no	—
22	C₂²×C₄×C₁₁	—	2	2	no	—
23	C₂²×C₄×C₁₁	—	2	2	no	—
24	C₂²×C₄×C₁₁	C₈	1	4	yes	complemented
25	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₁	D₄	1	2	yes	—
31	C₂²×C₄×C₁₁	Q₈	1	2	yes	—
32	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
33	C₄²×C₁₁	—	2	2	no	—
34	C₄²×C₁₁	—	2	2	no	—
35	C₄²×C₁₁	—	2	2	no	—
36	C₄²×C₁₁	D₄	1	4	yes	—
37	C₂×C₁₁⋊C₈	—	2	4	no	—
38	C₂×C₁₁⋊C₈	—	2	4	no	—
39	C₂×C₁₁⋊C₈	—	2	4	no	—
40	#1797	—	11	1	no	maximal, Sylow
41	C₂³×C₁₁	—	2	1	no	—
42	C₂³×C₁₁	—	2	1	no	—
43	C₂³×C₁₁	—	2	1	no	—
44	C₂³×C₁₁	—	2	1	no	—
45	C₂³×C₁₁	C₄⋊C₄	1	2	yes	—
46	C₂³×C₁₁	C₂×C₈	1	2	yes	—
47	C₂³×C₁₁	M₁₆	1	2	yes	—
48	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₁	—	2	8	no	—
50	C₂×C₄×C₁₁	—	2	1	no	—
51	C₂×C₄×C₁₁	—	2	1	no	—
52	C₂×C₄×C₁₁	—	2	1	no	—
53	C₂×C₄×C₁₁	—	2	1	no	—
54	C₂×C₄×C₁₁	—	2	1	no	—
55	C₂×C₄×C₁₁	—	2	1	no	—
56	C₂×C₄×C₁₁	—	2	1	no	—
57	C₂×C₄×C₁₁	—	2	1	no	—
58	C₂×C₄×C₁₁	—	2	8	no	—
59	C₂×C₄×C₁₁	—	2	4	no	—
60	C₂×C₄×C₁₁	—	2	4	no	—
61	C₂×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₁₁	C₂×C₈	1	2	yes	—
65	C₂×C₄×C₁₁	C₂×C₈	1	2	yes	—
66	C₂×C₄×C₁₁	M₁₆	1	2	yes	—
67	C₂×C₄×C₁₁	M₁₆	1	2	yes	—
68	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₁	#41	1	2	yes	—
72	C₁₁⋊C₈	—	4	4	no	—
73	C₂²×C₄²	D₁₁	1	1	yes	characteristic
74	C₂×C₄⋊C₈	—	11	2	no	—
75	C₂×C₂²⋊C₈	—	11	2	no	—
76	#336	—	11	2	no	—
77	C₂²×C₁₁	—	2	2	no	—
78	C₂²×C₁₁	—	2	2	no	—
79	C₂²×C₁₁	—	2	2	no	—
80	C₂²×C₁₁	—	2	2	no	—
81	C₂²×C₁₁	—	2	2	no	—
82	C₂²×C₁₁	—	2	2	no	—
83	C₂²×C₁₁	C₄⋊C₈	1	4	yes	complemented
84	C₂²×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₁₁	#117	1	1	yes	characteristic
87	C₂²×C₁₁	C₂²×C₈	1	1	yes	characteristic
88	C₂²×C₁₁	C₂×M₁₆	1	1	yes	characteristic
89	C₂²×C₁₁	C₂×M₁₆	1	1	yes	characteristic
90	C₂²×C₁₁	C₂×M₁₆	1	1	yes	characteristic
91	C₄×C₁₁	—	2	2	no	—
92	C₄×C₁₁	—	2	2	no	—
93	C₄×C₁₁	—	2	2	no	—
94	C₄×C₁₁	—	2	2	no	—
95	C₄×C₁₁	C₂²⋊C₈	1	4	yes	complemented
96	C₄×C₁₁	#121	1	2	yes	—
97	C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₄	Dic₁₁	1	1	yes	characteristic
99	C₂³×C₄	Dic₁₁	1	1	yes	characteristic
100	C₂³×C₄	C₂×D₁₁	1	1	yes	characteristic
101	C₂²×C₈	—	11	4	no	—
102	C₂×C₄²	—	2	4	no	—
103	C₂×C₄²	Dic₁₁	1	1	yes	characteristic
104	C₂×C₄²	Dic₁₁	1	1	yes	characteristic
105	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
106	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
107	C₄⋊C₈	—	22	4	no	—
108	C₂²⋊C₈	—	22	4	no	—
109	C₂×C₁₁	—	2	4	no	—
110	C₂×C₁₁	#380	1	1	yes	characteristic
111	C₂×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
112	C₂×C₁₁	#407	1	1	yes	characteristic
113	C₂×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₂×C₁₁	#423	1	1	yes	characteristic
115	C₂×C₁₁	#431	1	1	yes	characteristic
116	C₂×C₁₁	#432	1	1	yes	characteristic
117	C₂⁴	C₂×Dic₁₁	1	1	yes	characteristic
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	C₁₁⋊C₈	1	4	yes	complemented
126	C₂²×C₄	Dic₂₂	1	2	yes	—
127	C₂²×C₄	D₄₄	1	2	yes	—
128	C₂²×C₄	C₂×Dic₁₁	1	1	yes	characteristic
129	C₂²×C₄	C₂×Dic₁₁	1	1	yes	characteristic
130	C₂²×C₄	C₂×Dic₁₁	1	1	yes	characteristic
131	C₂²×C₄	C₂×Dic₁₁	1	1	yes	characteristic
132	C₂²×C₄	C₂×Dic₁₁	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
134	C₂×C₈	—	22	4	no	—
135	C₂×C₈	—	22	4	no	—
136	C₂×C₈	—	22	4	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	2	2	no	—
140	C₄²	C₂²⋊D₁₁	1	4	yes	—
141	C₁₁	#1797	1	1	yes	characteristic, complemented, Sylow
142	C₂³	—	2	1	no	—
143	C₂³	—	2	1	no	—
144	C₂³	—	2	1	no	—
145	C₂³	—	2	1	no	—
146	C₂³	C₂×C₁₁⋊C₈	1	2	yes	—
147	C₂³	#4318	1	2	yes	—
148	C₂³	C₄⋊Dic₁₁	1	2	yes	—
149	C₂³	C₂²×Dic₁₁	1	1	yes	characteristic
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
163	C₂×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
164	C₂×C₄	#4318	1	2	yes	—
165	C₂×C₄	#4318	1	2	yes	—
166	C₂×C₄	C₄⋊Dic₁₁	1	2	yes	—
167	C₂×C₄	C₂²⋊Dic₁₁	1	2	yes	—
168	C₂×C₄	C₂²⋊Dic₁₁	1	2	yes	—
169	C₂×C₄	C₂×Dic₂₂	1	1	yes	characteristic
170	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
171	C₂×C₄	#4339	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
173	C₈	—	44	4	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	C₄⋊(C₁₁⋊C₈)	1	4	yes	complemented
181	C₂²	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
182	C₂²	C₂×#4318	1	1	yes	characteristic
183	C₂²	C₂×#4318	1	1	yes	characteristic
184	C₂²	C₂×#4318	1	1	yes	characteristic
185	C₂²	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
186	C₂²	#66979	1	1	yes	characteristic
187	C₂²	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	C₂²⋊(C₁₁⋊C₈)	1	4	yes	complemented
193	C₄	#66977	1	2	yes	—
194	C₄	#66983	1	2	yes	—
195	C₂	—	2	4	no	—
196	C₂	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
197	C₂	#10622537	1	1	yes	characteristic
198	C₂	#10622538	1	1	yes	characteristic
199	C₂	#10622539	1	1	yes	characteristic
200	C₂	C₂×C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
201	C₂	#10622820	1	1	yes	characteristic
202	C₂	#10622825	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂²×C₈
Fitting	C₂²×C₄²×C₁₁	C₂
socle	C₂³×C₁₁	C₂²×C₄
prime radical	C₂⁴×C₁₁	C₂×C₄
cosocle	C₂²×C₄×C₁₁	C₂³
hypercenter	C₂²×C₄²	D₁₁
hypocenter	C₁₁	#1797
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×C₄⋊(C₁₁⋊C₈)	C₂	semidirect
4. #10622162	C₂	semidirect
24. C₂²×C₄×C₁₁	C₈	semidirect
83. C₂²×C₁₁	C₄⋊C₈	semidirect
95. C₄×C₁₁	C₂²⋊C₈	semidirect
125. C₂²×C₄	C₁₁⋊C₈	semidirect
141. C₁₁	#1797	semidirect
180. C₂²	C₄⋊(C₁₁⋊C₈)	semidirect
192. C₄	C₂²⋊(C₁₁⋊C₈)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄²×C₁₁	2
C₂×C₂²⋊(C₁₁⋊C₈)	2
C₂×C₄⋊(C₁₁⋊C₈)	2
#10622162	2
#1797	11